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Question

Solve the following equations which have equal roots:
x4(a+b)x3a(ab)x2+a2(a+b)xa3b=0.

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Solution

Give equation, x4(a+b)x3a(ab)x2+a2(a+b)xa3b=0

Consider f(x)=x4(a+b)x3a(ab)x2+a2(a+b)xa3b

f(x)=4x33(a+b)x22a(ab)x+a2(a+b)

Now, HCF of f(x) and f(x) is (xa). Hence x=a is a double root of f(x)=0

f(x) can be factored as (xa)2(x2+(ab)xab)

roots of the given equation are a,a,b,a

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